Note that the scaling by
in (B.46) is necessary to
maintain unit area under each impulse.

We will now show that

(B.47)

That is, the Fourier transform of the normalized impulse train
is exactly the same impulse train
in the
frequency domain, where
denotes time in seconds and
denotes
frequency in Hz. By the scaling theorem (§B.4),

Thus, the
-periodic impulse train transforms to a
-periodic
impulse train, in which each impulse contains area
:

(B.49)

Proof:
Let's set up a limiting construction by defining

(B.50)

so that
. We may interpret
as a sampled rectangular pulse of width
seconds (yielding
samples).
By linearity of the Fourier transform and the shift
theorem (§B.5), we readily obtain the transform of
to be

where we have used the definition of
given in
Eq.(3.5) of §3.1. As we would
expect from basic sampling theory, the Fourier transform of the
sampled rectangular pulse is an aliased sinc function.
Figure 3.2 illustrates one period
for
.

The proof can be completed by expressing the aliased sinc function as
a sum of regular sinc functions, and using linearity of the Fourier
transform to distribute
over the sum, converting each sinc
function into an impulse, in the limit, by §B.13:

and verify that the peaks occur every
seconds and reach height
. Also show that the peak widths, measured between zero
crossings, are
, so that the area under each peak is of
order 1 in the limit as
. [Hint: The shift theorem for
inverse Fourier transforms is
, and
.]